用户: Scavenger/Prismatic Cohomology

Fix a prime number $p$ throughout.

Convention 0.1. We use the language of $\infty$-category throughout. By “$\infty$-category” we mean $(\infty ,1)$-categories. When we are talking about $\infty$-categories we always use the term “$\infty$-category”, so in particular the term “category” always means ordinary $1$-categories.

We always use homological index convention when talking about chain complexes and related objects such as differential granded algebra, even if we are using superscripts.

We often use the letter “d” (which is short for the word “discrete”) to indicate that we are working in the $1$-categorical or classical context. For example, if $A$ is an ordinary ring, we use ${\mathrm{dMod}}_A$ to mean the $1$-category of discrete (or classical) modules over $A$, and leave the notation ${\mathrm{Mod}}_A$ to denote the $\infty$-category of $A$-module spectra. If $M,N$ are two discrete $A$-modules, then we use $M{\otimes }_A^{d}N$ to denote the classical tensor product, and leave the notation $M{\otimes }_{A}N$ to denote the tensor product of $M$ and $N$ as $A$-module spectra.

As another example, let $A$ be an ordinary ring, $I\subseteq A$ a finitely generated ideal, and $M$ a discrete $A$-module. We can form the derived $I$-completion of $M$ either in the category of discrete $A$-modules (so the result is forced to be discrete) or in the $\infty$-category of $A$-module spectra (so the result is not necessarily discrete). We will refer to the first situation as discrete derived completion and the second situation as the nondiscrete derived completion.

1Recasting Sheaf Cohomology Using $\infty$-Categorical Techniques

Definition 1.1. Let $\mathcal{T}$ be an $\infty$-category. A Grothendieck topology (or simply a topology) on $\mathcal{T}$ is a Grothendieck topology on the homotopy category $\text{h}\mathcal{T}$ (see [7] Subsection 6.2.2 for related discussions).

Let $\mathcal{T}$ be an $\infty$-category equipped with a topology. Let $\mathcal{C}$ be an $\infty$-category. We say that a functor $F:{\mathcal{T}}^{\mathrm{op}}\to \mathcal{C}$ is a $\mathcal{C}$-valued sheaf on $\mathcal{T}$ if the following condition is satisfied: For every object $U\in \mathcal{T}$ and every covering sieve ${\mathcal{T}}_0\subseteq {\mathcal{T}}_{/U}$ on $U$, the composite map ${\mathcal{T}}_0^{▹}\to ({\mathcal{T}}_{/U}{)}^{▹}\to \mathcal{T}\stackrel{F^{\mathrm{op}}}{\to }{\mathcal{C}}^{\mathrm{op}}$ is a colimit diagram in ${\mathcal{C}}^{\mathrm{op}}$. We let ${\mathrm{Sh}}_{\mathcal{C}}(\mathcal{T})$ denote the full subcategory of $\mathrm{Fun}({\mathcal{T}}^{\mathrm{op}},\mathcal{C})$ spanned by the $\mathcal{C}$-valued sheaves on $\mathcal{T}$.

Example 1.2. Let $\mathcal{T}$ be a small $\infty$-category equipped with a topology. Let $\mathcal{C}$ be a presentable $\infty$-category. Then the full subcategory ${\mathrm{Sh}}_{\mathcal{C}}(\mathcal{T})\subseteq \mathrm{Fun}({\mathcal{T}}^{\mathrm{op}},\mathcal{C})$ is a reflective full subcategory. The localization functor corresponding to this reflective full subcategory is called the sheafification functor.

Example 1.3 ([8] Example 1.3.1.3). Let $\mathcal{T}$ be a small $\infty$-category equipped with a topology, let $\mathcal{S}$ be the $\infty$-category of spaces. We will denote the $\infty$-category ${\mathrm{Sh}}_{\mathcal{S}}(\mathcal{T})$ simply by $\mathrm{Sh}(\mathcal{T})$, and refer to it as the $\infty$-category of sheaves on $\mathcal{T}$. One can check that the definition of sheaves here coincides with the one of [7] Definition 6.2.2.6. In particular $\mathrm{Sh}(\mathcal{T})$ is an $\infty$-topos.

Definition 1.4. Let $\mathcal{C}$ and $\mathcal{D}$ be small $\infty$-categories equipped with Grothendieck topologies. A functor $F:\mathcal{C}\to \mathcal{D}$ is called cocontinuous if for every object $C\in \mathcal{C}$ and every covering sieve ${\mathcal{D}}_0\subseteq {\mathcal{D}}_{/F(C)}$ on $F(C)$, ${\mathcal{D}}_0{\times }_{{\mathcal{D}}_{/F(C)}}{\mathcal{C}}_{/C}\subseteq {\mathcal{C}}_{/C}$ is a covering sieve on $C$.

Construction 1.5 (Derived Pushforward of Sheaves). Let $\mathcal{C}$ and $\mathcal{D}$ be small $\infty$-categories equipped with Grothendieck topologies. Let $\mathcal{X}$ be a presentable $\infty$-category. Let $f:\mathcal{C}\to \mathcal{D}$ be a cocontinuous functor. Then there is a pair of adjoint functors $f_{\ast }:{\mathrm{Sh}}_{\mathcal{X}}(\mathcal{C})\begin{array}{c}\leftarrow \\ \to \end{array}{\mathrm{Sh}}_{\mathcal{X}}(\mathcal{D}):f^{\ast }$ (where $f^{\ast }:{\mathrm{Sh}}_{\mathcal{X}}(\mathcal{D})\to {\mathrm{Sh}}_{\mathcal{X}}(\mathcal{C})$ is the left adjoint and $f_{\ast }:{\mathrm{Sh}}_{\mathcal{X}}(\mathcal{C})\to {\mathrm{Sh}}_{\mathcal{X}}(\mathcal{D})$ is the right adjoint) constructed as follows: $f^{\ast }$ is given by precomposing with $f^{\mathrm{op}}$ followed by sheafification, and $f_{\ast }$ is given by right Kan extension along $f^{\mathrm{op}}$ (so in particular the right Kan extension of a $\mathcal{X}$-valued sheaf on $\mathcal{C}$ along $f^{\mathrm{op}}$ is a $\mathcal{X}$-valued sheaf on $\mathcal{D}$). $f^{\ast }$ and $f_{\ast }$ should be the analogues of derived pullback and derived pushforward in the context of $\infty$-categories. When taking $\mathcal{X}$ to be the $\infty$-category of spectra or the $\infty$-category of $E_{\infty }$-rings we obtain scenario analogous to the derived pushforward of abelian sheaves or sheaves of rings on sites.

Example 1.6. Let $\mathcal{C}$ be an $\infty$-category equipped with a Grothendieck topology, let $\mathcal{X}$ be a presentable $\infty$-category. Let $\ast$ be the final $\infty$-category, that is, the $1$-category with only one object and one morphism. Then the unique functor $\mathcal{C}\to \ast$ is automatically cocontinuous, and the derived pushforward along this functor, as constructed in Construction 1.5, should be the analogue of the usual functor $R\Gamma ()$.

Definition 1.7 ([8] Definition 1.3.1.4). Let $\mathcal{X}$ be an $\infty$-topos, let $\mathcal{C}$ be an $\infty$-category. A $\mathcal{C}$-valued sheaf on $\mathcal{X}$ is a functor ${\mathcal{X}}^{\mathrm{op}}\to \mathcal{C}$ that preserves small limits. Let ${\mathrm{Sh}}_{\mathcal{C}}(\mathcal{X})$ denote the full subcategory of $\mathrm{Fun}({\mathcal{X}}^{\mathrm{op}},\mathcal{C})$ spanned by the $\mathcal{C}$-valued sheaves on $\mathcal{X}$.

Proposition 1.8 ([8] Proposition 1.3.1.7). Let $\mathcal{T}$ be a small $\infty$-category equipped with a topology. Let $j:\mathcal{T}\to P(\mathcal{T})$ denote the Yoneda embedding, let $L:P(\mathcal{T})\to \mathrm{Sh}(\mathcal{T})$ be a left adjoint to the inclusion $\mathrm{Sh}(\mathcal{T})\to P(\mathcal{T})$. Let $\mathcal{C}$ be an arbitrary $\infty$-category which admits small limits. Then composition with $L\circ j$ induces an equivalence of $\infty$-categories ${\mathrm{Sh}}_{\mathcal{C}}(\mathrm{Sh}(\mathcal{T}))\to {\mathrm{Sh}}_{\mathcal{C}}(\mathcal{T})$.

Proposition 1.9 ([8] Lemma 1.3.4.3). Let $\mathcal{X}$ be an $\infty$-topos and $\mathcal{C}$ a presentable $\infty$-category. Then the inclusion ${\mathrm{Sh}}_{\mathcal{C}}(\mathcal{X})\to \mathrm{Fun}({\mathcal{X}}^{\mathrm{op}},\mathcal{C})$ admits a left adjoint. This left adjoint functor is called the sheafification functor.

Example 1.10. Let $\mathcal{C}$ and $\mathcal{D}$ be small $\infty$-categories equipped with Grothendieck topologies. Let $\mathcal{X}$ be a presentable $\infty$-category. Let $f:\mathcal{C}\to \mathcal{D}$ be a cocontinuous functor. $(f^{\ast },f_{\ast }):\mathrm{Sh}(\mathcal{C})\begin{array}{c}\leftarrow \\ \to \end{array}\mathrm{Sh}(\mathcal{D})$ gives a geometric morphism between $\infty$-topoi, and we thus have a pair of adjoint functors $f^{\ast }:{\mathrm{Sh}}_{\mathcal{X}}(\mathrm{Sh}(\mathcal{C}))\begin{array}{c}\leftarrow \\ \to \end{array}{\mathrm{Sh}}_{\mathcal{X}}(\mathrm{Sh}(\mathcal{D})):f_{\ast }$, where $f^{\ast }$ is the left adjoint and $f_{\ast }$ is the right adjoint, constructed as follows: $f^{\ast }$ is given by precomposing with $f_{\ast }^{\mathrm{op}}:\mathrm{Sh}(C{)}^{\mathrm{op}}\to \mathrm{Sh}(D{)}^{\mathrm{op}}$ followed by the sheafification functor, and $f_{\ast }$ is given by precomposing with $f^{\ast }:\mathrm{Sh}(D{)}^{\mathrm{op}}\to \mathrm{Sh}(C{)}^{\mathrm{op}}$. One can check that this adjoint pair coincides with that of Construction 1.5 under the identification given by Proposition 1.8.

Weakly Final Objects and Cech Resolution

We will only be considering $\infty$-categories equipped with the indiscrete topology for convenience, but the technique surely applies in more general context.

Example 1.11 (Weakly Final Objects and Cech Resolution). Let $\mathcal{C}$ be a small $\infty$-category. Let $F:{\mathcal{C}}^{\mathrm{op}}\to \mathcal{S}$ be a presheaf on $\mathcal{C}$, let $G:{\mathcal{C}}^{\mathrm{op}}\to \ast \to \mathcal{S}$ be the final presheaf on $\mathcal{C}$. Then $F\to G$ is an effective epimorphism in $P(\mathcal{C})$ if and only if for every object $C\in \mathcal{C}$, the space $F(C)$ is nonempty, by [7] Corollary 6.2.3.5.

Let $X\in \mathcal{C}$ be an object. It is said to be weakly final if $h_X\to G$ is an effective epimorphism in $P(\mathcal{C})$, where $h_X$ is the presheaf represented by $X$. Then $X$ is weakly final if and only if for every object $Y\in \mathcal{C}$, ${\mathrm{Hom}}_{\text{h}\mathcal{C}}(Y,X)$ is nonempty.

Let $X\in \mathcal{C}$ be a weakly final object, let $U_{+}:{\Delta }_{+}^{\mathrm{op}}\to P(\mathcal{C})$ be the Cech nerve of $h_X\to G$, then $U_{+}$ is a colimit diagram in $P(\mathcal{C})$. Let $U:U_{+}{|}_{{\Delta }^{\mathrm{op}}}$ denote the underlying simplicial object of $U_{+}$. If $\mathcal{C}$ admits finite nonempty products, then $U$ factors through the Yoneda embedding $\mathcal{C}\to P(\mathcal{C})$.

Let $\mathcal{X}$ be a presentable $\infty$-category and let $\mathcal{F}:P(\mathcal{C}{)}^{\mathrm{op}}\to \mathcal{X}$ be an $\mathcal{X}$-valued sheaf on $P(\mathcal{C})$. Then the composite functor ${\Delta }_{+}\stackrel{U^{\mathrm{op}}}{\to }P(\mathcal{C}{)}^{\mathrm{op}}\stackrel{\mathcal{F}}{\to }\mathcal{X}$ is a limit diagram in $\mathcal{X}$. If $\mathcal{C}$ admits nonempty finite products, then this functor exhibits $\mathcal{F}(G)$ as the totalization (i.e. limit) of a cosimplicial object in the essential image of the composite functor ${\mathcal{C}}^{\mathrm{op}}\to P(\mathcal{C}{)}^{\mathrm{op}}\stackrel{\mathcal{F}}{\to }\mathrm{CAlg}$, where the first map is the Yoneda embedding.

Using Example 1.10 we see that $R\Gamma (\mathcal{F})$ (in the sense of Example 1.6) is isomorphic to $\mathcal{F}(G)$. Thus we can compute $R\Gamma (\mathcal{F})=\mathcal{F}(G)$ as the totalization of a cosimplicial object, which we call the Cech resolution.

2Preliminaries on Commutative Algebra

Definition 2.1. Let $A$ be an ordinary ring, let $I\subseteq A$ be a finitely generated ideal. An $A$-module spectra $M$ is called $I$-completely flat if $M{\otimes }_{A}A/I$ is a flat discrete $A/I$-module. If $M$ is an $I$-completely flat $A$-module spectra, then so is its nondiscrete derived $I$-completion.

An $A$-module spectra $M$ is said to have finite $I$-complete Tor amplitude if $M{\otimes }_{A}A/I$ has finite Tor amplitude in $\mathcal{D}(A/I)$.

An $A$-module spectra $M$ is called $I$-completely faithfully flat if $M{\otimes }_{A}A/I$ is a faithfully flat discrete $A/I$-module.

Let $B$ be an ordinary $A$-algebra. It is said to be $I$-completely smooth ($I$-completely etale) over $A$ if it is derived $I$-complete and $B{\otimes }_{A}A/I$ is a discrete smooth (etale) $A/I$-algebra.

Proposition 2.2. Let $A\to B$ be a map of $E_{\infty }$-rings, let $I\subseteq {\pi }_{0}A$ be a finitely generated ideal, let $J:=I\cdot {\pi }_{0}B\subseteq {\pi }_{0}B$. Let $M$ be a $B$-module spectrum. Then $M$ is $J$-nilpotent ($J$-local, $J$-complete) if and only if it is $I$-nilpotent ($I$-local, $I$-complete) when being regarded as an $A$-module spectrum.

Proof. The nilpotent case is simple. Let’s consider the local case. Let $f_1,...,f_n$ be a set of generators of $I$. Let $V_i=\mathrm{fib}(A\to A[1/f_i])$ in ${\mathrm{Mod}}_A$, and let $V={⨂}_{1\le i\le n}{V}_i$ in ${\mathrm{Mod}}_A$. Then an $A$-module spectrum $N$ is $I$-local if and only if $V{\otimes }_{A}N=0$. The same construction with $(A,I)$ replaced by $(B,J)$ yields $V^{\prime }=V{\otimes }_{A}B$, and we see that $M{\otimes }_B{V}^{\prime }=M{\otimes }_{A}V$, so the former vanishes precisely when the latter vanishes.

Now we turn to the complete case. An $A$-module spectrum $N$ is $I$-complete if and only if ${\mathrm{Ext}}_A^k(A[1/f_i],N)=0$ for all $k\in \mathbb{Z}$ and $1\le i\le n$. Since ${\mathrm{Ext}}_B^k(B[1/f_i],M)\simeq {\mathrm{Ext}}_B^k(A[1/f_i]{\otimes }_{A}B,M)\simeq {\mathrm{Ext}}_A^k(A[1/f_i],M)$, so the former vanishes precisely when the latter vanishes.

$\square$

2.3. Let $A$ be an ordinary ring and let $I\subseteq A$ be a finitely ideal. Choose a finite set $f_1,...,f_n$ of generators of $I$. Consider the sequence $...\to \mathrm{fib}(\mathrm{Kos}(A;f_1^2,...,f_n^2)\to A/(f_1^2,...,f_n^2))\to \mathrm{fib}(\mathrm{Kos}(A;f_1^1,...,f_n^1)\to A/(f_1^1,...,f_n^1))$ of $A$-module spectra. We assume that this sequence is essentially zero. That is, for every $i\in {\mathbb{Z}}_{\ge 1}$, there exists a $j\in {\mathbb{Z}}_{>i}$ such that the map from the $j$th element to the $i$th element in the above sequence is zero. Let $M$ be an $A$-module spectrum, then tensoring the above sequence with $M$, the resulting sequence is again essentially zero, and consequently its limit is zero. Thus the $I$-completion of $M$ is given by $\lim (...\to M{\otimes }_{A}A/(f_1^2,...,f_n^2)\to M{\otimes }_{A}A/(f_1^1,...,f_n^1))$. If we assume $M$ is flat over $A$ (and in particular $M$ is discrete), then $\lim (...\to M{\otimes }_{A}A/(f_1^2,...,f_n^2)\to M{\otimes }_{A}A/(f_1^1,...,f_n^1))=\lim (...\to M/(f_1^2,...,f_n^2)M\to M/(f_1^1,...,f_n^1)M)$, which is discrete as one can compute, and equals the classical $I$-completion of $M$.

Proposition 2.4. Let $A$ be a $\delta$-ring, $I\subseteq A$ a finitely generated ideal containing $p$. Let $B$ be an ordinary $A$-algebra that is a discrete derived $I$-completion of an ordinary etale $A$-algebra (this condition holds in particular when $B$ is $I$-completely etale). Then $B$ admits a unique $\delta$-structure compatible with the one on $A$.

Proof. We use the characterization of $\delta$-structures in terms of $W_2$. Let $B^{\prime }$ be an etale $A$-algebra, and $B^{\prime }\to B$ a map of $A$-algebras that exhibits $B$ as the discrete derived $I$-completion of $B^{\prime }$. We have a commutative diagram of short exact sequences of discrete $A$-modules

where ${\varphi }_{\ast }{B}^{\prime }$ and ${\varphi }_{\ast }B$ are $B^{\prime }$ and $B$ regarded as discrete $A$-modules through the ring maps $A\stackrel{\varphi }{\to }A\to B^{\prime }$ and $A\stackrel{\varphi }{\to }A\to B$. Since $p\in I$, $\varphi (I)A$ and $I$ have the same radical, thus the discrete derived $I$-completion of ${\varphi }_{\ast }{B}^{\prime }$ is ${\varphi }_{\ast }B$. Since the left and right vertical arrows in the above diagram are discrete derived $I$-completion arrows, so is the middle one.

Consider the commutative diagram

where the left vertical arrow is etale. Since $W_2(B)$ is derived $(p)$-complete, if we let $f:=(0,1)\in W_2(B)$, then $W_2(B)$ is $(f)$-complete. Therefore $W_2(B)\to W_2(B)/(f)$ and hence $W_2(B)\to B$ is a transfinite composition of square zero extensions in $\mathrm{AniRing}$, the $\infty$-category of animated rings, and consequently $W_2(B)\to B$ is right orthogonal to $A\to B^{\prime }$ in $\mathrm{AniRing}$, and there exists a unique map $B^{\prime }\to W_2(B)$ rendering the diagram

commutative. This map then extends to a unique map $B\to W_2(B)$ by the universal property of discrete derived completion, and we are done.

$\square$

From the proof of Proposition 2.4 we obtain the following

Corollary 2.5. Let $A$ be a $\delta$-ring, $I\subseteq A$ a finitely generated ideal containing $p$. Let $B$ be the discrete derived $I$-completion of $A$, equipped with the $\delta$-structure provided by Proposition 2.4. Then for every $\delta$-ring $C$ over $A$ such that $C$ is derived $I$-complete, ${\mathrm{Hom}}_{\delta \text{-Ring}}(B,C)\to {\mathrm{Hom}}_{\delta \text{-Ring}}(A,C)$ is a bijection.

More generally, let $A$ be a $\delta$-ring, $I\subseteq A$ a finitely generated ideal containing $p$. Let $B$ be an ordinary etale $A$-algebra, let $B^{\prime }$ be the discrete derived $I$-completion of the $A$-algebra $B$, equipped with the $\delta$-structure provided by Proposition 2.4. Let $C$ be a $\delta$-ring over $A$ such that $C$ is derived $I$-complete, let $B\to C$ be an $A$-algebra map. Then the $A$-algebra map $B^{\prime }\to C$ provided by the universal property of discrete derived completion is a map of $\delta$-rings.

3Prisms

Definition 3.1. A $\delta$-pair is a pair $(A,I)$ where $A$ is a $\delta$-ring and $I\subseteq A$ is an ideal. A morphism between two $\delta$-pairs $(A,I)$ and $(B,J)$ is a morphism $f:A\to B$ of $\delta$-rings such that $f(I)\subseteq J$.

A prism is a $\delta$-pair $(A,I)$ satisfying the following conditions:

(i) $I$ is a Cartier divisor, which means that the quasicoherent sheaf on $\mathrm{Spec}A$ determined by the discrete $A$-module $I$ is a line bundle.

(ii) $A$ is derived $(p,I)$-complete as a module spectrum over itself.

(iii) $p\in I+\varphi (I)A$.

A morphism of prisms is a morphism of the underlying $\delta$-pairs.

A prism $(A,I)$ is bounded if $A/I$ has bounded $p^{\infty }$-torsion.

A prism $(A,I)$ is crystalline if $I=(p)$.

A prism $(A,I)$ is orientable if $I$ is principal. An orientation for an orientable prism $(A,I)$ is a choice of a generator of $I$. An oriented prism is an orientable prism together with an orientation.

Notation 3.2. Let $(A,I)$ be a prism. The ring $\overline{A}:=A/I$ is called the slice of $(A,I)$.

Proposition 3.3 (Rigidity of Prisms). Let $(A,I)\to (B,J)$ be a morphism of prisms. Then the natural map $I{\otimes }_A^{d}B\to J$ is an isomorphism of discrete $B$-modules. In particular $J=IB$.

Proof. See [5] Lemma 5.4.2.

$\square$

3.4. Let $(A,I)$ be a bounded prism. Then the ideal $(p,I)\subseteq A$ satisfies the hypothesis of 2.3, where we use the “set of generators” $\{p^n,I\}$ for some $n\in {\mathbb{Z}}_{\ge 1}$ sufficiently large (here we use the cartier divisor $I$ directly in the Koszul complexes, instead of choosing a set of generators of $I$). Consequently we have the following

Proposition 3.5. Let $(A,I)$ be a bounded prism, let $M$ be a flat discrete $A$-module. Let $\hat{M}$ be the classical $(p,I)$-completion of $M$. Then the map $M\to \hat{M}$ exhibits $\hat{M}$ as the nondiscrete derived $(p,I)$-completion of $M$. In particular the nondiscrete derived $(p,I)$-completion of $M$ is discrete, and $A$ is itself classically $(p,I)$-complete.

Similar things hold if we replace $A$ by $\overline{A}$ and $(p,I)$ by $(p)$.

Example 3.6. Let $(A,I)$ be a bounded prism. Let $R$ be an ordinary $\overline{A}$-algebra. Then $R$ is a $(p)$-completely smooth (etale) $\overline{A}$-algebra if and only if $R$ is classically $(p)$-complete and the map $\mathrm{Spf}(R)\to \mathrm{Spf}(\overline{A})$ (where both $R$ and $\overline{A}$ are equipped with the $(p)$-adic topology) is a smooth (etale) map of formal schemes (or more concretely, for every $n\in {\mathbb{Z}}_{\ge 1}$, the map $\overline{A}/p^n\overline{A}\to R/p^{n}R$ is smooth (etale)).

Proof. We only treat the smooth case. The etale case is similar. If $R$ is $(p)$-completely smooth over $\overline{A}$, then by Elkik’s algebraization theorem $R$ is the nondiscrete derived $(p)$-completion of an ordinary smooth $\overline{A}$-algebra, hence also the classical $(p)$-completion of this $\overline{A}$-algebra by Proposition 3.5, and consequently $\mathrm{Spf}(R)\to \mathrm{Spf}(\overline{A})$ is smooth.

If $\mathrm{Spf}(R)\to \mathrm{Spf}(\overline{A})$ is smooth, then using the same proof as Elkik’s algebraization theorem, we see that $R$ is the classical $(p)$-completion of an ordinary smooth $\overline{A}$-algebra, thus it is $(p)$-completely smooth over $\overline{A}$.

$\square$

Proposition 3.7. Let $(A,I)$ be a bounded prism. Let $M$ be a derived $(p,I)$-complete and $(p,I)$-completely flat $A$-module spectrum. Then $M$ is discrete and classically $(p,I)$-complete. For any $n\in {\mathbb{Z}}_{\ge 1}$ we have $M[I^n]=0$ and $M/I^{n}M$ has bounded $p^{\infty }$-torsion.

Remark 3.8. A similar thing holds for $\overline{A}$. See [2] Lemma 4.7.

Proof. See [3] Lemma 3.7 (2). Notice that there misses the derived completeness condition in the statement there.

$\square$

Corollary 3.9. Let $(A,I)\to (A^{\prime },I{A}^{\prime })$ be a map of bounded prisms. Let $(B,J)$ be a $\delta$-pair over $(A,I)$ such that $B$ is derived $(p,I)$-complete and $(p,I)$-completely flat over $A$. Then $(B{\otimes }_A{A}^{\prime }{)}^{\wedge }$ ( $({)}^{\wedge }$ here means nondiscrete derived $(p,I)$-completion) is derived $(p,I)$-complete and $(p,I)$-completely flat over $A^{\prime }$ and in particular discrete. Notice that $((B{\otimes }_A{A}^{\prime }{)}^{\wedge },J((B{\otimes }_A{A}^{\prime }{)}^{\wedge }))$ is the pushout of the diagram

in the category of derived $(p,I)$-complete $\delta$-pairs over $(A,I)$ (notice that $(B{\otimes }_A{A}^{\prime }{)}^{\wedge }$ is the discrete derived $(p,I)$-completion of $B{\otimes }_A^d{A}^{\prime }$, so in particular admits a natural $\delta$-structure), so in particular if $(B,J)$ is a bounded prism over $(A,I)$ then $((B{\otimes }_A{A}^{\prime }{)}^{\wedge },J((B{\otimes }_A{A}^{\prime }{)}^{\wedge }))$ is the pushout of the above diagram in the category of bounded prisms.

4The Prismatic Site

Construction 4.1. Let $(A,I)$ be a prism with slice $\overline{A}$. Let $R$ be an $\overline{A}$-algebra. The prismatic site of $R$ relative to $A$, denoted $(R/A{)}_{\mathbf{\Delta }}$, is defined to be the opposite category of the following category, where:

(i) An object consists of a morphism of prisms $(A,I)\to (B,IB)$ together with a morphism of $\overline{A}$-algebras $R\to B/IB$. We will typically notate such an object as $(R\to B/IB\leftarrow B)$ and depict such an object as a diagram below:

(ii) A morphism between two objects $(R\to B/IB\leftarrow B)$ and $(R\to C/IC\leftarrow C)$ will be a morphism of prisms $(B,IC)\to (C,IC)$ compatible with the above diagram.

We will always equip $(R/A{)}_{\mathbf{\Delta }}$ with the indiscrete Grothendieck topology (and in particular every presheaf is a sheaf).

Example 4.2. Let $(A,I)$ be a bounded prism and $R$ a $p$-completely smooth $A/I$-algebra. Then there exists a prism $(B,J)$ over $(A,I)$ such that $B/IB\simeq R$ as $A/I$-algebras.

Proof. Use Elkik’s algebraization theorem we choose a smooth $A/I$-algebra $R^{\prime }$ and a map $R^{\prime }\to R$ of $A/I$-algerbas that exhibits $R$ as the nondiscrete derived $(p)$-completion of $R^{\prime }$. Use [11] tag 07M8 to choose a smooth $A$-algebra $S^{\prime }$ such that $S^{\prime }/I{S}^{\prime }\simeq R^{\prime }$ as $A/I$-algebras. Let $S^{\prime }\to S$ be a map of $A$-algebras that exhibits $S$ as the nondiscrete derived $(p,I)$-completion of $S^{\prime }$. By Proposition 3.5 $S$ is discrete and equal to the classical $(p,I)$-completion of $S^{\prime }$. Then $\mathrm{fib}(S^{\prime }\to S)$ is $(p,I)$-local, thus $\mathrm{fib}(S^{\prime }\to S){\otimes }_{A}A/I$ is $p$-local. Since $A$ and $I$ are invertible discrete modules over $A$, tensoring with them does not affect $(p,I)$-nilpotence/locality/completeness, so $S{\otimes }_{A}A/I\simeq \mathrm{cofib}(S{\otimes }_{A}I\to S{\otimes }_{A}A)$ is $(p)$-complete. So $S{\otimes }_{A}A/I$ is the nondiscrete derived $(p)$-completion of $S^{\prime }{\otimes }_{A}A/I\simeq R^{\prime }$, and in particular $S{\otimes }_{A}A/I$ is discrete and isomorphic to $R$ as $A/I$-algebras.

Now using the fact that, for any ordinary ring $B$, a $\delta$-structure on $B$ is equivalent to a section of the projection $W_2(B)\to B$, we can give $S^{\prime }$ a $\delta$-structure compatible with the one of $A$. Using [3] Lemma 2.17 we can give $S$ a $\delta$-structure compatible with the one of $S^{\prime }$. We claim that $(S,IS)$ is a prism. The only nontriviality is to prove that $IS$ is a Cartier divisor. Since $S{\otimes }_{A}A/I$ is discrete, $S{\otimes }_A^{d}I\to S$ is injective, thus $IS\simeq S{\otimes }_A^{d}I$ is a line bundle over $S$. The prism $(S,IS)$ is the prism we are looking for.

$\square$

Construction 4.3 (Prismatic and Hodge-Tate Cohomology). Let $(A,I)$ be a prism with slice $\overline{A}$. Let $R$ be an $\overline{A}$-algebra. We define sheaves of rings ${\mathcal{O}}_{\mathbf{\Delta }}$ and ${\overline{\mathcal{O}}}_{\mathbf{\Delta }}$ on $(R/A{)}_{\mathbf{\Delta }}$ as follows: ${\mathcal{O}}_{\mathbf{\Delta }}((R\to B/IB\leftarrow B))=B$ and ${\overline{\mathcal{O}}}_{\mathbf{\Delta }}((R\to B/IB\leftarrow B))=B/IB$.

The (unique) functor $p:(R/A{)}_{\mathbf{\Delta }}\to \ast$ (where $\ast$ is the category with one object and one morphism) is cocontinuous, thus we can preform pushforwards ($p_{\ast }$) and pullbacks ($p^{\ast }$) along $p$, and $(p^{\ast },p_{\ast })$ is an adjoint pair). In particular we have a functor $p_{\ast }:{\mathrm{Sh}}_{E_{\infty }\text{-Ring}}((R/A{)}_{\mathbf{\Delta }})\to E_{\infty }\text{-Ring}$, where $E_{\infty }\text{-Ring}$ is the $\infty$-category of $E_{\infty }$-rings. Define $E_{\infty }$-rings ${\mathbf{\Delta }}_{R/A}:=p_{\ast }({\mathcal{O}}_{\mathbf{\Delta }})$ and ${\overline{\mathbf{\Delta }}}_{R/A}:=p_{\ast }({\overline{\mathcal{O}}}_{\mathbf{\Delta }})$.

Notice that the pullback functor $p^{\ast }:E_{\infty }\text{-Ring}\to {\mathrm{Sh}}_{E_{\infty }\text{-Ring}}((R/A{)}_{\mathbf{\Delta }})$ is given by composition with $p:(R/A{)}_{\mathbf{\Delta }}\to \ast$, so $p^{\ast }(B)=B$ (the constant sheaf on $B$) for every $E_{\infty }$-ring $B$. Using adjointness we see that we have a natural diagram of $E_{\infty }$-rings:

The natural frobenius endormorphisms $\varphi$ of prisms induces an endormorphism $\varphi :{\mathcal{O}}_{\mathbf{\Delta }}\to {\mathcal{O}}_{\mathbf{\Delta }}$ of the sheaf of rings ${\mathcal{O}}_{\mathbf{\Delta }}$, which is compatible with the frobenius endormorphism of $A$. Applying $p_{\ast }$ (and using adjunction) we see that ${\mathbf{\Delta }}_{R/A}$ admits a natural endormorphism $\varphi$ compatible with the frobenius endormorphism of $A$.

Proposition 4.4. The diagram

is a pushout diagram in $E_{\infty }\text{-Ring}$.

Proof. For any sheaf of $A$-module spectra $M$ on $(R/A{)}_{\mathbf{\Delta }}$ and an $A$-module spectrum $N$, we have a natural map $p_{\ast }M{\otimes }_{A}N\to p_{\ast }(M{\otimes }_A{p}^{\ast }N)$ induced by adjunction, thus functorial in $M$ and $N$. For $N=A$ or $N=I$ this map is easily checked to be an isomorphism, thus this map is an isomorphism for $N=A/I=\mathrm{cofib}(I\to A)$. Taking $M={\mathcal{O}}_{\mathbf{\Delta }}$ we get what we want (notice that ${\overline{\mathcal{O}}}_{\mathbf{\Delta }}={\mathcal{O}}_{\mathbf{\Delta }}{\otimes }_{A}A/I$).

$\square$

Proposition 4.5. ${\mathbf{\Delta }}_{R/A}$ is a $(p,I)$-complete $A$-module spectrum, and consequently ${\overline{\mathbf{\Delta }}}_{R/A}$ is a $(p)$-complete $\overline{A}$-module spectrum.

Proof. For $k\in \mathbb{Z}$ and $x\in (p,I)\subseteq A$, ${\mathrm{Hom}}_{{\mathrm{Mod}}_A}((A[1/x])[k],{\mathbf{\Delta }}_{R/A})\simeq {\mathrm{Hom}}_{{\mathrm{Sh}}_{{\mathrm{Mod}}_A}((R/A{)}_{\mathbf{\Delta }})}((A[1/x])[k],{\mathcal{O}}_{\mathbf{\Delta }})$, so we only need to prove that the latter is contractible. Recall that ${\mathcal{O}}_{\mathbf{\Delta }}:(R/A{)}_{\mathbf{\Delta }}^{\mathrm{op}}\to {\mathrm{Mod}}_A$ is a functor, thus ${\mathrm{Hom}}_{{\mathrm{Sh}}_{{\mathrm{Mod}}_A}((R/A{)}_{\mathbf{\Delta }})}((A[1/x])[k],{\mathcal{O}}_{\mathbf{\Delta }})\simeq {\mathrm{Hom}}_{{\mathrm{Mod}}_A}((A[1/x])[k],\lim ({\mathcal{O}}_{\mathbf{\Delta }}))$ is contractible since $\lim ({\mathcal{O}}_{\mathbf{\Delta }})$ is $(p,I)$-complete.

$\square$

Lemma 4.6. Let $(A,I)$ be a prism. Then the forgetful functor from the category of prisms over $(A,I)$ to the category of $\delta$-pairs over $(A,I)$ admits a left adjoint, which we call the prismatic envelope.

Proof. Let $(B,J)$ be a $\delta$-pair over $(A,I)$. We construct the prismatic envelope for $(B,J)$. For every $\delta$-ring $C$ and discrete $C$-module $M$, there exists a $\delta$-ring $C\{M\}$ over $C$ such that for every $\delta$-ring $X$ over $C$, there exists a bijection from ${\mathrm{Hom}}_{\delta {\text{-Ring}}_{C/}}(C\{M\},X)$ to ${\mathrm{Hom}}_{C\text{-Mod}}(M,X)$, functorial in $X$. Let $F\subseteq I^{-1}{\otimes }_A^{d}B$ denote the subset consisting of elements $x$ such that for every element $i\in I$, $ix\in I{\otimes }_A^d{I}^{-1}{\otimes }_A^{d}B\cong B$ lies in $J$, then $F$ is a sub discrete $B$-module (one can show that $F=I^{-1}{\otimes }_A^{d}J$). Consider the $\delta$-ring $B\{F\}$ together with the universal map $j:F\to B\{F\}$. For every pair of elements $i\in I$ and $f\in F$, consider the element $q(i,f):=l(i)\cdot j(f)-l(i\cdot f)\in B\{F\}$, where $l:B\to B\{F\}$. Let $B^{\prime }$ be the $\delta$-ring obtained form $B\{F\}$ by quotient out elements of the form $q(i,f)$ (that is, $B^{\prime }$ is $B\{F\}$ quotient out the minimum ideal stable under the $\delta$ operation and contains $q(i,f)$, as ordinary rings). Then the map $(B,J)\to (B^{\prime },I{B}^{\prime })$ is a map of $\delta$-pairs. Moreover, if $(C,IC)$ is a prism over $(A,I)$, then the map ${\mathrm{Hom}}_{\delta \text{-Pair}}((B^{\prime },I{B}^{\prime }),(C,IC))\to {\mathrm{Hom}}_{\delta \text{-Pair}}((B,J),(C,IC))$ is a bijection. Summarizing what we’ve done above, we see that we may assume $J=IB$.

Let $B_0:=B$, for every ordinal $i$, let $B_{i+1}^{\prime }$ be the maximum $I$-torsion free quotient of $B_i$ (taken in the context of $\delta$-rings, so $B_{i+1}^{\prime }$ admits a natural $\delta$-structure) (one can easily show that such a maximum $I$-torsion free quotient indeed exists), and let $B_{i+1}$ be the discrete derived $(p,I)$-completion of $B_{i+1}^{\prime }$ (so $B_{i+1}$ naturally admits a $\delta$-structure). Let $B_i={\mathrm{colim}}_{j<i}(B_j)$ (colimit taken in the category of $\delta$-rings) if $i$ is a nonzero limit ordinal. Since the category of derived $(p,I)$-complete discrete $A$-modules is stable under $\kappa$-filtered $1$-colimits, where $\kappa >\omega$ is a regular cardinal, the map $(B,IB)\to (B_{\kappa },I{B}_{\kappa })$ is the prismatic envelope we are looking for.

$\square$

Lemma 4.7. Let $(A,I)$ be a prism, let $R$ be an $\overline{A}$-algebra. Then the category $(R/A{)}_{\mathbf{\Delta }}$ admits finite nonempty products.

Proof. See the proof of [5] Lemma 11.6.4.

$\square$

Proposition 4.8. Let $(A,I)$ be a prism, let $R$ be an $\overline{A}$-algebra. Then the category $(R/A{)}_{\mathbf{\Delta }}$ admits a weakly final object.

Proof. The proof of the corresponding proposition (Proposition 11.6.5) of [5] is wrong. We give a correct proof below.

Let $B_0:=A[R]$ be the polynomial $A$-algebra on the set of free generators $R$, so we have a surjection $B_0\to R$ of $A$-algebras, mapping a free generator $r\in R$ to $r\in R$. Let $B$ be the free $\delta$-ring over $A$ generated by the set $R$, so we have a natural map $B_0\to B$. Let $J=\ker (B_0\to R)$, then one can show that the prismatic envelope of $(B,JB)$ gives a weakly final object.

$\square$

Remark 4.9. Since the prismatic site admits a weakly final object, we can use Example 1.11 to compute the cohomology of sheaves on the prismatic site by choosing a weakly final object and then the cohomology is obtained as the totalization of the Cech resolution associated to this weakly final object. We will be implicitly using this computation throughout our proof of the Hodge-Tate comparison theorem.

Proposition 4.10. Let $(A,I)$ be a bounded prism, and let $B$ be a $(p,I)$-completely flat $\delta$-$A$-algebra. Let $J\subseteq B$ be an ideal containing $IB$, so that there is an induced map $(A,I)\to (B,J)$ of $\delta$-pairs. Assume that Zariski locally on $\mathrm{Spf}(B)$ we can write $J=(I,x_1,...,x_r)$ for a sequence $x_1,...,x_r\in B$ that is $(p,I)$-completely regular relative to $A$ (in the sense of [3] Definition 2.42).

Let $(C,IC)$ be the prismatic envelope of $(B,J)$ relative to $(A,I)$. Then:

(i) $C$ is $(p,I)$-completely flat over $A$.

(ii) Let $(A,I)\to (A^{\prime },I{A}^{\prime })$ be a map of bounded prisms (so in particular $(A^{\prime },I{A}^{\prime })$ is bounded), then $((C{\otimes }_A{A}^{\prime }{)}^{\wedge },I((C{\otimes }_A{A}^{\prime }{)}^{\wedge }))$ ($({)}^{\wedge }$ here means nondiscrete derived $(p,I)$-completion) is the prismatic envelope of $(B{\otimes }_A^d{A}^{\prime },J(B{\otimes }_A^d{A}^{\prime }))$ relative to $(A^{\prime },I{A}^{\prime })$ (see Corollary 3.9 for justification for this statement).

(iii) The prismatic envelope of $(B,J)$ is compatible with flat localization on B, which means that if $B\to B^{\prime }$ is a $(p,I)$-completely flat map of $\delta$-$A$-algebras, let $(C^{\prime },I{C}^{\prime })$ be the prismatic envelope of $(B^{\prime },J{B}^{\prime })$ over $(A,I)$, so we have a commutative diagram

of $\delta$-pairs, then the natural map $C{\otimes }_B{B}^{\prime }\to C^{\prime }$ exhibits $C^{\prime }$ as the nondiscrete derived $(p,I)$-completion of $C{\otimes }_B{B}^{\prime }$.

Proof. See [3] Proposition 3.13.

$\square$

Example 4.11. This is Example 3.14 of [3]. A proof doesn’t seem to be given there and I don’t know how to prove this statement, so I’ll just copy it verbatim and pretend it is correct.

Let $(A,I)$ be a bounded prism, let $R$ be an ordinary $(p)$-completely smooth $A/I$-algebra. Let $B_0$ be an ordinary $(p,I)$-completely smooth $A$-algebra and let $B_0\to R$ be a surjection of ordinary $A$-algebras. Let $B$ be a $\delta$-$A$-algebra and let $B_0\to B$ be a $(p,I)$-completely flat map of ordinary $A$-algebras. Let $J_0\subseteq B_0$ be the kernel of $B_0\to R$, $R^{\prime }$ be the discrete derived $(p)$-completion of $B/J_{0}B$ and let $J\subseteq B$ be the kernel of $B\to R^{\prime }$. Then $(B,J)$ satisfies the condition of Proposition 4.10. (Notice that the prismatic envelope of $(B,J)$ is the same as the prismatic envelope of $(B,J_{0}B)$.)

4.12. Let $(A,I)$ be a bounded prism. Let $R$ be an ordinary $(p)$-completely smooth $\overline{A}$-algebra. Using Example 4.11 (I think) [3] Construction 4.17 and Construction 4.18 claims that the category $(R/A{)}_{\mathbf{\Delta }}$ admits a weakly final object $(R\to B/IB\leftarrow B)$ such that $(B,IB)$ is a bounded prism, and the elements of the Cech nerve associated to this weakly final object are all bounded prisms.

(In the constructions we will encounter ind-smooth algebras instead of smooth algebras, so a filtered colimit procedure is required. I don’t know the exact way to perform this filtered colimit procedure (and I think this has not beed mentioned in [3]), so I’ll just pretend that there will not be any problems.)

The prismatic site in [3] is defined using bounded prisms ([3] Definition 4.1), instead of allowing all not necessarily bounded prisms, which is the definition we’ve been using. The above claim shows that this discrepancy doesn’t really matter: both definitions of the prismatic site give the same cohomology in the case we really care about.

5The Hodge-Tate Comparison Map

5.1 (Totalization). I don’t know how to prove the following facts, but I think they are correct:

Let $A:\Delta \to \mathrm{Ab}$ be a cosimplicial abelian group,where $\mathrm{Ab}$ is the category of abelian groups, then (I think) [4] Problem 4.23 shows that $\lim A$ (limit taken in the derived $\infty$-category $\mathcal{D}(\mathbb{Z})$) is naturally equivalent to the complex associated to the cosimplicial abelian group $A$ whose differentials are given by alternating sums, as in [5] Definition 11.2.2.

The above discussion naturally extends to the case of cosimplicial chain complexes of abelian groups. More precisely, the limit functor $\mathrm{Fun}(\Delta ,\mathrm{Ch}(\mathbb{Z}))\to \mathrm{Fun}(\Delta ,\mathcal{D}(\mathbb{Z}))\stackrel{\lim }{\to }\mathcal{D}(\mathbb{Z})$ is equivalent to the functor $\mathrm{Fun}(\Delta ,\mathrm{Ch}(\mathbb{Z}))\stackrel{f}{\to }\mathrm{Ch}(\mathbb{Z})\to \mathcal{D}(\mathbb{Z})$, where $f$ is the functor first taking a cosimplicial chain complex to the underlying double chain complex, and then to the totalization of this double chain complex. Moreover, I think this equivalence should be compatible with the projection from the limit of a cosimplicial chain complex to the level $0$ chain complex of the cosimplicial chain complex.

Now assume $A:\Delta \to \mathrm{dRing}$ is a cosimplicial commutative ring, where $\mathrm{dRing}$ is the category of ordinary commutative rings. I think the monoidal Dold-Kan correspondence equips the associated complex of abelian groups of $A$ the structure of an $E_{\infty }$-ring (see this nLab page), and this should be the limit of $A$ in $E_{\infty }\text{-Ring}$, the $\infty$-category of $E_{\infty }$-rings. In particular the commutative graded ring structure on the homotopy groups is given by the cup product (see this nLab page), that is, for $a\in A([i])$, $b\in A([j])$, both are cycles of the complex, we have $a\cdot b=(x_{\ast }(a)\in A([i+j]))\cdot (y_{\ast }(b)\in A([i+j]))$, where $x:[i]\to [i+j]$ is the inclusion of the first $i+1$ elements, and $y:[j]\to [i+j]$ is the inclusion of the last $j+1$ elements.

Definition 5.2. Let $A\to B$ be a morphism of ordinary rings. Define the de Rham complex to be the cdga (commutative differential graded algebra) over $A$ given by the chain complex $B\to {\Omega }_{B/A}^1\to {\Omega }_{B/A}^2\to ...$, where ${\Omega }_{B/A}^i={\wedge }_B^i{\Omega }_{B/A}$, whose multiplication is given by the wedge product.

The de Rham complex of $B$ over $A$ will be denoted $({\Omega }_{B/A}^{\bullet },d_{\mathrm{dR}})$, or simply ${\Omega }_{B/A}^{\bullet }$.

Lemma 5.3 (Universal Property for de Rham Complex). Let $A\to B$ be a morphism of ordinary rings. Let $(E^{\bullet },d)$ be a coconnective cdga over $A$. Let $\eta :B\to E^0$ be a map of $A$-algebras such that for each element $x\in B$, $y=d(\eta (x))\in E^{-1}$ satisfies $y^2=0$. Then $\eta$ extends uniquely to a map ${\Omega }_{B/A}^{\bullet }\to E^{\bullet }$ of cdgas over $A$.

Proof. See [5] Lemma 12.2.3.

$\square$

Definition 5.4. Let $A$ be an ordinary ring and $I\subseteq A$ a finitely generated ideal. The subcategory of the category of chain complexes of discrete $A$-modules spanned by the chain complexes of discrete $A$-modules that are levelwise derived $I$-complete is a reflective subcategory, which is compatible with the symmetric monoidal structure in the sense of [6] Definition 2.2.1.6. Consequently the full subcategory of the category of $A$-cdgas spanned by the levelwise derived $I$-complete $A$-cdgas is a reflective subcategory, and the localization functor associated to this reflective subcategory is given by levelwise discrete derived $I$-completion.

The derived $I$-completion of the de Rham complex will be called the completed de Rham complex, denoted ${\hat{\Omega }}_{B/A}^{\bullet }$. It satisfies a similar universal property as that of ordinary de Rham complex.

Computation 5.5. Let $(A,I)$ be a bounded prism. Let $R$ be a $(p)$-completely smooth $\overline{A}$-algebra, so there exists a smooth $\overline{A}$-algebra $R^{\prime }$ and a map $R^{\prime }\to R$ of $\overline{A}$-algebras that exhibits $R$ as the nondiscrete derived $(p)$-completion of $R^{\prime }$. Let $i\in {\mathbb{Z}}_{\ge 1}$. In this case we have a canonical map $f:{\Omega }_{R^{\prime }/\overline{A}}^i{\otimes }_{R^{\prime }}^{d}R\to {\Omega }_{R/\overline{A}}^i$. Since $R^{\prime }$ is flat over $\overline{A}$ and $(A,I)$ is bounded, $R$ is the classical $(p)$-completion of $R^{\prime }$, thus ${\Omega }_{R^{\prime }/\overline{A}}^i{\otimes }_{R^{\prime }}^{d}R$ is the classical $(p)$-completion of ${\Omega }_{R^{\prime }/\overline{A}}^i$ since ${\Omega }_{R^{\prime }/\overline{A}}^i$ is finite projective over $R^{\prime }$. Notice that $f$ becomes an isomorphism after modulo $p^n$, where $n\in {\mathbb{Z}}_{\ge 1}$, so we have a map $g:{\Omega }_{R/\overline{A}}^i\to {\Omega }_{R^{\prime }/\overline{A}}^i{\otimes }_{R^{\prime }}^{d}R$ such that $g\circ f=\mathrm{id}$, so ${\Omega }_{R/\overline{A}}^i=({\Omega }_{R^{\prime }/\overline{A}}^i{\otimes }_{R^{\prime }}^{d}R)\oplus M$, where $M$ is a discrete $R$-module. It is easy to check that $pM=M$, so the discrete derived $(p)$-completion of $M$ vanishes. Therefore the discrete derived $(p)$-completion of ${\Omega }_{R/\overline{A}}^i$ is equal to the classical $(p)$-completion, which is also equal to ${\Omega }_{R^{\prime }/\overline{A}}^i{\otimes }_{R^{\prime }}^{d}R$. Summarizing what we’ve done so far, we see that ${\hat{\Omega }}_{R/\overline{A}}^{\bullet }\simeq {\Omega }_{R^{\prime }/\overline{A}}^{\bullet }{\otimes }_{R^{\prime }}^{d}R$ (the completion is the levelwise classical $(p)$-completion here), and in particular each term of it is finite projective over $R$.

Construction 5.6. Let $(A,I)$ be a prism, let $R$ be an $\overline{A}$-algebra. For $M$ a discrete $\overline{A}$-module and $n\in \mathbb{Z}$, define the Breuil-Kisin twist $M\{n\}:=M{\otimes }_{\overline{A}}^d(I/I^2{)}^{{\otimes }^{d}n}$.

Let $M$ be an $A$-module spectrum. For $n\in \mathbb{Z}$, tensoring the fiber sequence $I^{n+1}/I^{n+2}\to I^n/I^{n+2}\to I^n/I^{n+1}$ in ${\mathrm{Mod}}_A$ with $M$ we obtain a fiber sequence $M{\otimes }_A{I}^{n+1}/I^{n+2}\to M{\otimes }_A{I}^n/I^{n+2}\to M{\otimes }_A{I}^n/I^{n+1}$. Considering the connecting homomorphism we obtain a map ${\beta }^n:{\pi }_{-n}(M{\otimes }_A{I}^n/I^{n+1})\to {\pi }_{-n-1}(M{\otimes }_A{I}^{n+1}/I^{n+2})$ of discrete $A$-modules. These maps gives the ${\pi }_{-n}(M{\otimes }_A{I}^n/I^{n+1})$’s the structure of a chain complex of discrete $A$-modules by [5] Lemma 12.3.2. Moreover, if $M$ can be promoted to an $E_{\infty }$-ring over $A$, then this chain complex has a natural structure of an $A$-cdga (the Lebniz rule can be easily seen by observing that $A/I^2\to A/I$ is a square zero extension of animated rings by the $A/I$-module $I/I^2$).

In particular letting $M={\mathbf{\Delta }}_{R/A}$ we get a cdga structure on ${\pi }_{-n}({\overline{\mathbf{\Delta }}}_{R/A})\{n\}$.

Proposition 5.7. Let $\eta :R\to {\pi }_0({\overline{\mathbf{\Delta }}}_{R/A})$ be the map induced by $\eta :R\to {\overline{\mathbf{\Delta }}}_{R/A}$. For any element $t\in R$, ${\beta }^1(\eta (t))\in {\pi }_{-1}({\overline{\mathbf{\Delta }}}_{R/A})\{1\}$ squares to zero in ${\pi }_{-2}({\overline{\mathbf{\Delta }}}_{R/A})\{2\}$.

A proof of the above proposition, implicitly using the unproved claim 5.1, is given in [5] Lemma 12.3.4 (there are some calculation errors that don’t matter essentially). For the readers feeling unsafe about this unproved claim, there is always a functorial approach, see the comment after Lemma 5.4 of [1] Lecture 5, at the top of page 8.

Using the universal property of completed de Rham complex, we get a map of cdgas ${\hat{\Omega }}_{R/\overline{A}}^{\bullet }\to {\pi }_{-\bullet }({\overline{\mathbf{\Delta }}}_{R/A})\{\bullet \}$ (the completion is the $(p)$-completion here). This map is called the Hodge-Tate comparison map.

Now we can formulate the Hodge-Tate comparison theorem:

Theorem 5.8 (Hodge-Tate Comparison Theorem). Let $(A,I)$ be a bounded prism. Let $R$ be a $(p)$-completely smooth $\overline{A}$-algebra. Then the Hodge-Tate comparison map constructed above is an isomorphism of cdgas.

6Proof of the Hodge-Tate Comparison Theorem

Definition 6.1 (Divided Power Envelope). Let $R$ be an ordinary ring that is flat over $\mathbb{Z}$ (or equivalently $R$ is $n$-torsion free for $n\in {\mathbb{Z}}_{\ne 0}$). The divided power operations ${\gamma }_n:R\to R{\otimes }_{\mathbb{Z}}^d\mathbb{Q}$ are the maps of sets defined by the formula ${\gamma }_n(x)=\frac{x^n}{n!},x\in R,n\in {\mathbb{Z}}_{\ge 0}$.

Let $J$ be an ideal of $R$. The divided power envelope of (R,J) is the subring $D$ of $R{\otimes }_{\mathbb{Z}}^d\mathbb{Q}$ generated by $R$ and ${\gamma }_n(x)$ for all $x\in J$ and $n\in {\mathbb{Z}}_{\ge 0}$. We also denote this ring by $D_J(R)$. If $\{x_i{\}}_{i\in I}$ is a set of generators of the ideal $J$, then $D_J(R)$ is equal to the subring of $R{\otimes }_{\mathbb{Z}}^d\mathbb{Q}$ generated by $R$ and ${\gamma }_n(x_i)$ for all $i\in I$ and $n\in {\mathbb{Z}}_{\ge 0}$.

Aside 6.2 (Cosimplicial Objects). Let $\mathcal{C}$ be an ordinary category. The category $\mathrm{Fun}(\Delta ,\mathcal{C})$ is called the category of cosimplicial objects of $\mathcal{C}$ (here $\Delta$ is the simplex category), and its elements are called cosimplicial objects of $\mathcal{C}$.

If $\mathcal{C}$ admits finite products, then $\mathrm{Fun}(\Delta ,\mathcal{C})$ admits a canonical simplicial enrichment. In this case for $X$ a finite simplicial set and $A$ a cosimplicial object of $\mathcal{C}$, we define a new cosimplicial object of $\mathcal{C}$, denoted $A^X$, by the formula $A^X([n])={\prod }_{X_n}A([n])$, with the obvious transition maps. Now for $A,B$ two cosimplicial objects of $\mathcal{C}$ we define a simplicial set $\mathrm{Hom}(A,B)$ by the formula $\mathrm{Hom}(A,B{)}_n={\mathrm{Hom}}_{\mathrm{Fun}(\Delta ,\mathcal{C})}(A,B^{{\Delta }^n})$. It is easy to check that this gives $\mathrm{Fun}(\Delta ,\mathcal{C})$ a simplicial enrichment. Moreover, if $\mathcal{D}$ is another ordinary category which admits finite products, and $\mathcal{C}\to \mathcal{D}$ is a functor preserving finite products, then the induced functor $\mathrm{Fun}(\Delta ,\mathcal{C})\to \mathrm{Fun}(\Delta ,\mathcal{D})$ can be naturally promoted into a simplicially enriched functor.

Let $\mathcal{C}$ be a category which admits finite products. The simplicial category $\mathrm{Fun}(\Delta ,\mathcal{C})$ has a natural homotopy category, which we denote by $\mathrm{hFun}(\Delta ,\mathcal{C})$, obtained by taking ${\pi }_0$ at each $\mathrm{Hom}$-simplicial set of $\mathrm{Fun}(\Delta ,\mathcal{C})$. Let $A,B$ be two cosimplicial objects of $\mathcal{C}$, and let $f,g:A\to B$ be two morphisms of cosimplicial objects of $\mathcal{C}$. $f$ and $g$ are called homotopic if they define the same morphism in $\mathrm{hFun}(\Delta ,\mathcal{C})$. $f$ is said to be a homotopy equivalence if it defines an equivalence in $\mathrm{hFun}(\Delta ,\mathcal{C})$. Let $\mathcal{D}$ be a category which admits finite products and let $F:\mathcal{C}\to \mathcal{D}$ be a functor preserving finite products, then $F:\mathrm{Fun}(\Delta ,\mathcal{C})\to \mathrm{Fun}(\Delta ,\mathcal{D})$ preserves homotopic morphisms and homotopy equivalences.

Remark 6.3. The notions of homotopic morphisms and homotopy equivalences between cosimplicial objects of a category $\mathcal{C}$ can actually be defined even if the $\mathcal{C}$ is not assumed to have finite products, and these notions are preserved by an arbitrary functor $F:\mathcal{C}\to \mathcal{D}$ between categories, even if $F$ is not assumed to preserve finite products. See [11] tag 0FKJ for details.

Let $\mathrm{dMod}$ be the $1$-category of pairs $(A,M)$ where $A$ is an ordinary ring and $M$ is a discrete $A$-module, so we a natural forgetful functor $\mathrm{dMod}\to \mathrm{dRing}$, where $\mathrm{dRing}$ is the category of ordinary rings, mapping a pair $(A,M)$ to the ring $A$. The induced functor $\mathrm{Fun}(\Delta ,\mathrm{dMod})\to \mathrm{Fun}(\Delta ,\mathrm{dRing})$ can be naturally promoted to a simplicially enriched functor. Let $A$ be a cosimplicial ring, we let ${\mathrm{Mod}}_A:=\mathrm{Fun}(\Delta ,\mathrm{dMod}){\times }_{\mathrm{Fun}(\Delta ,\mathrm{dRing})}\{A\}$, which has a natural simplicial enrichement, and consequently a corresponding theory of homotopic morphisms and homotopy equivalences. Objects of ${\mathrm{Mod}}_A$ will be called cosimplicial $A$-modules. We have a natural tensor product operation in ${\mathrm{Mod}}_A$, which is formed by levelwise tensor product. Let $f,f^{\prime }:M\to M^{\prime }$ be two homotopic morphisms of cosimplicial $A$-modules, let $N$ be a cosimplicial $A$-module, then it is easy to show that $f\otimes N$ and $f^{\prime }\otimes N$ are homotopic.

Let $\mathcal{A}$ be an abelian category. For a cosimplicial object $A$ of $\mathcal{A}$ we have the associated Moore complex $s(A)$ (see [11] tag 019H for definitions), which is a chain complex concerntrated in nonpositive degrees, so in particular $s(A{)}_{-n}=A([n])$ for $n\in {\mathbb{Z}}_{\ge 0}$. A morphism $A\to B$ of cosimplicial objects of $\mathcal{A}$ is called a weak equivalence if $s(A)\to s(B)$ is a quasi-isomorphism. It is easy to check that the functor $s:\mathrm{Fun}(\Delta ,\mathcal{A})\to {\mathrm{Ch}}_{\le 0}(\mathcal{A})$ descends to a functor $s:\mathrm{hFun}(\Delta ,\mathcal{A})\to {\mathrm{hCh}}_{\le 0}(\mathcal{A})$, where ${\mathrm{hCh}}_{\le 0}(\mathcal{A})$ is the homotopy category of nonpositively graded chain complexes (see [11] tag 01A0). In particular, homotopy equivalences between cosimplicial objects of $\mathcal{A}$ are weak equivalences.

Proposition 6.4 (Cech Nerves). Let $f:X\to Y$ be a morphism of a category $\mathcal{C}$ which admits pushouts. Assume that there is a morphism $s:Y\to X$ such that $s\circ f={\mathrm{id}}_X$. Consider the Cech nerve $U$ (or better, opposite Cech nerve, see [11] tag 016N for an explicit construction) of $f$ (this is a cosimplicial object in $\mathcal{C}$). The morphism $U\to U$ which in each degree is the self map of $Y{\coprod }_X...{\coprod }_{X}Y$ given by $f\circ s$ on each factor is homotopic to ${\mathrm{id}}_U$. In particular $U$ is homotopy equivalent to the constant cosimplicial object on $X$.

Proof. See [11] tag 019Z.

$\square$

Lemma 6.5 ([3] Lemma 5.4). Let $k$ be an ordinary commutative ring of characteristic $p$. Let $B$ be a polynomial $k$-algebra. Let $B^{\bullet }$ be the (opposite) Cech nerve of $k\to B$, and let $B^{\bullet ,(1)}$ be its Frobenius twist (given by (levelwise) pushing out along the Frobenius endomorphism of $k$), so we have the relative Frobenius $B^{\bullet ,(1)}\to B^{\bullet }$ as a map of cosimplicial $k$-algebras. Then for any cosimplicial discrete $B^{\bullet ,(1)}$-module $N$, the natural map $N\to N{\otimes }_{B^{\bullet ,(1)}}^d{B}^{\bullet }$ (the tensor product here is given by levelwise tensor product) is a weak equivalence of cosimplicial discrete $k$-modules.

Notation 6.6. Let $A$ be a $\delta$-ring. Let $J\subseteq A$ be an ideal, we let $J_{\delta }$ be the minimum ideal of $A$ containing $J$ and stable under $\delta$. Let $x_1,...,x_n,d\in A$ be elements (where $n\in {\mathbb{Z}}_{\ge 0}$). We use $A\{\frac{x_1,...,x_n}{d}\}$ to denote the $\delta$-ring $A\{t_1,...,t_n\}/(d{t}_1-x_1,...,d{t}_n-x_n{)}_{\delta }$.

Proposition 6.7. Let $m\le n\in {\mathbb{Z}}_{\ge 1}$. Let $B:={\mathbb{Z}}_p\{x_1,...,x_n\}$ be the free $\delta$-ring generated by the free variables $x_1,...,x_n$. Let $J\subseteq B$ be the ideal generated by $x_1,...,x_m$. Then there is a canonical isomorphism of rings $B\{\frac{\varphi (x_i),1\le i\le m}{p}\}\cong D_J(B)$.

Proof. By definition, $B\{\frac{\varphi (x_i),1\le i\le m}{p}\}={\mathbb{Z}}_p\{x_1,...,x_n,t_1,...,t_m\}/(p{t}_1-\varphi (x_1),...,p{t}_m-\varphi (x_m){)}_{\delta }$. By the proof of [5] Lemma 14.3.2, $D_J(B)$ is a sub $\delta$-ring of $B{\otimes }_{\mathbb{Z}}^d\mathbb{Q}\cong {\mathbb{Q}}_p\{x_1,...,x_n\}$. Consider the map of $\delta$-rings $B\{\frac{\varphi (x_i),1\le i\le m}{p}\}\to D_J(B)$, mapping $t_i$ ($1\le i\le m$) to $\frac{\varphi (x_i)}{p}$ and $x_j$ ( $1\le j\le n$) to $x_j$. One can show that this map is an isomorphism (see [1] Lecture 6, Lemma 2.1, [5] Corollary 14.3.3 and [3] Lemma 2.36).

$\square$

Aside 6.8. See this mathoverflow page for discussions around the question of how to get an $E_{\infty }$-ring from a cdga. The most direct reason for this is the fact that $\mathrm{Ch}(\mathbb{Z})\to \mathrm{Ch}(\mathbb{Z})[w^{-1}]$ is lax symmetric monoidal, see [9] Theorem A.7. There is also another explanation for this construction, as in [10] Example 3.3.14:

Let $k$ be an ordinary commutative ring. Then the symmetric monoidal $\infty$-category of completely filtered k-module spectra has a compatible $t$-structure, called the Beilinson $t$-structure, whose heart coincides with the $1$-category of cochain complexes. The compatibility implies that the inclusion of the heart has a lax symmetric monoidal structure, thus a cdga gives rise to a filtered $E_{\infty }$-$k$-algebra. Passing to the underlying $E_{\infty }$-algebra, we get what we want.

Computation 6.9 (Crystalline-de Rham Comparison for Affine Spaces). Let $r,s\in {\mathbb{Z}}_{\ge 0}$. Let $R={\mathbb{F}}_p[x_1,...,x_r]$, $P_0={\mathbb{Z}}_p[x_1,...,x_r]$, $P={\mathbb{Z}}_p[x_1,...,x_r,y_1,...,y_s]$. Let $D$ be the divided power envelope of $(P,(y_1,...,y_s))$. Since $D$ is a subring of $P{\otimes }_{{\mathbb{Z}}_p}^d{\mathbb{Q}}_p$, one can naturally form a cdga $D{\otimes }_P^d{\Omega }_{P/Z_p}^{\bullet }$. Then one can show that the map ${\Omega }_{P_0/Z_p}^{\bullet }\to D{\otimes }_P^d{\Omega }_{P/Z_p}^{\bullet }$ is a quasi-isomorphism of cdgas by reducing to the case where $r=0,s=1$ and then conclude using explicit calculation (notice that the cdgas we are considering are repeated tensor products of cdgas corresponding to the case $(r,s)=(0,1)\text{ or }(1,0)$).

Using the same argument as above one can show that the map ${\Omega }_{P_0/Z_p}^{\bullet }\to D{\otimes }_P^d{\Omega }_{P/Z_p}^{\bullet }$ remains a quasi-isomorphism after levelwise modulo $p$, hence also remains a quasi-isomorphism after levelwise classical $(p)$-completion by derived Nakayama lemma ([11] tag 0G1U) since both ${\Omega }_{P_0/Z_p}^{\bullet }$ and $D{\otimes }_P^d{\Omega }_{P/Z_p}^{\bullet }$ are $p$-torsion free.

Computation 6.10 (Hodge-Tate Comparison Theorem for Affine Spaces). In the context of Construction 5.6, let $(A,I)=({\mathbb{Z}}_p,(p))$, $R={\mathbb{F}}_p[x_1,...,x_n]$ (where $n\in {\mathbb{Z}}_{\ge 1}$). Let $B:={\mathbb{Z}}_p\{x_1,...,x_n\}$ be the free $\delta$-ring generated by the free variables $x_1,...,x_n$, let $B^{\wedge }$ be the classical $(p)$-completion of $B$. Then it is easy to see that $R\cong B^{\wedge }/p{B}^{\wedge }$ and $(B^{\wedge },p{B}^{\wedge })$ is a weakly final object of $(R/A{)}_{\mathbf{\Delta }}$. We now give an explicit formula for the Cech nerve of $(A,I)\to (B^{\wedge },p{B}^{\wedge })$ in $(R/A{)}_{\mathbf{\Delta }}^{\mathrm{op}}$.

Let $B^{\bullet }$ be the Cech nerve of $A\to B$ formed in the category of $\delta$-rings, so $B^i=A\{x_j^{(k)},0\le k\le i,1\le j\le n\}$. One can check that $C^i:=B^i\{\frac{x_j^{(k)}-x_j^{(0)},1\le k\le i,1\le j\le n}{p}\}$ is a free $\delta$-ring over ${\mathbb{Z}}_p$ and in particular $p$-torsion free. Therefore the underlying cosimplicial ring of the Cech nerve of $(A,I)\to (B^{\wedge },p{B}^{\wedge })$ in $(R/A{)}_{\mathbf{\Delta }}^{\mathrm{op}}$ is (isomorphic to) $(C^{\bullet }{)}^{\wedge }$, where $(C^i{)}^{\wedge }$ is the classical $(p)$-completion of $C^i$.

One can show that there is a canonical isomorphism of rings $C^i{\otimes }_{B^i,\varphi }^d{B}^i\cong B^i\{\frac{\varphi (x_j^{(k)}-x_j^{(0)}),1\le k\le i,1\le j\le n}{p}\}=:D^i$, where the first ring is the pushout of $C^i$ along the Frobenius endomorphism of $B^i$. We show that $f:(C^{\bullet }{)}^{\wedge }\to (D^{\bullet }{)}^{\wedge }$ is a weak equivalence of cosimplicial rings (where the superscript $\wedge$ means classical $p$-completion). By virtue of Lemma 6.5, $f$ is a weak equivalence after quotient out $p$. As $(C^{\bullet }{)}^{\wedge }$ and $(D^{\bullet }{)}^{\wedge }$ are levelwise $p$-torsion free ($D^{\bullet }$ is $p$-torsion free since it can be identified with the divided power envelope of a free $\delta$-ring over ${\mathbb{Z}}_p$, by Proposition 6.7. $C^{\bullet }$ is $p$-torsion free since it is a free $\delta$-ring over ${\mathbb{Z}}_p$), we see that their mod $p$ reductions equal their derived mod $p$ reduction (i.e. derived tensoring with ${\otimes }_{\mathbb{Z}}\mathbb{Z}/p\mathbb{Z}$), so the totalization (i.e. taking limit of the corresponding cosimplicial objects in the $\infty$-category of $E_{\infty }$-rings or other suitable $\infty$-categories) of $f$ (regarded as a morphism in $\mathcal{D}(\mathbb{Z})$) is a quasi-isomorphism after derived mod $p$ reduction. Since the totalizations of $(C^{\bullet }{)}^{\wedge }$ and $(D^{\bullet }{)}^{\wedge }$ are limits of derived $(p)$-complete discrete modules, they are derived $(p)$-complete when being regarded as objects of $\mathcal{D}(\mathbb{Z})$, so by derived Nakayama lemma ([11] tag 0G1U), the totalization of $f$ is an equivalence in $\mathcal{D}(\mathbb{Z})$, and consequently $f$ is a weak equivalence of cosimplicial rings.

Let $P^{\bullet }$ be the cosimplicial ring whose level $i$ ($i\in {\mathbb{Z}}_{\ge 1}$) ring $P^i$ is ${\mathbb{Z}}_p[x_j^{(k)},0\le k\le i,1\le j\le n]$. Let $Q^{\bullet }$ be the cosimplicial ring whose level $i$ ring $Q^i$ is ${\mathbb{Z}}_p[x_j^{(k,l)},0\le k\le i,1\le j\le n,l\in {\mathbb{Z}}_{\ge 1}]$ (one should think $x_j^{(k,l)}$ as denoting the $l$-fold $\delta$ operation of $x_j^{(k)}$ in $B^i$). Let $J_i\subseteq P^i$ be the ideal generated by $x_j^{(k)}-x_j^{(0)},1\le k\le i,1\le j\le n$. One easily sees by Proposition 6.7 that the cosimplicial ring $D^{\bullet }$ is isomorphic to the cosimplicial ring $D_{J_{\bullet }}(P^{\bullet }){\otimes }_{{\mathbb{Z}}_p}{Q}^{\bullet }$. By [11] tag 023M the map ${\mathbb{Z}}_p\to Q^{\bullet }$ of cosimplicial rings is a weak equivalence, so consequently $D_{J_{\bullet }}(P^{\bullet })\to D_{J_{\bullet }}(P^{\bullet }){\otimes }_{{\mathbb{Z}}_p}{Q}^{\bullet }\cong D^{\bullet }$ is also a weak equivalence of cosimplicial rings (the cosimplicial ring $D_{J_{\bullet }}(P^{\bullet }){\otimes }_{{\mathbb{Z}}_p}{Q}^{\bullet }$ is the diagonal of the bicosimplicial ring (that is, a functor $\Delta \times \Delta \to \mathrm{dRing}$) $D_{J_{\bullet }}(P^{\bullet }){\otimes }_{{\mathbb{Z}}_p}{Q}^{\ast }$, so their limits agree. Notice the limit of the latter bicosimplicial ring when regarded as a bicosimplicial discrete ${\mathbb{Z}}_p$-module is the totalization of the underlying double chain complex). The same argument as above shows that $D_{J_{\bullet }}(P^{\bullet })/p{D}_{J_{\bullet }}(P^{\bullet })\to D^{\bullet }/p{D}^{\bullet }$ is a weak equivalence, so using derived Nakayama lemma ([11] tag 0G1U) we see that $D_{J_{\bullet }}(P^{\bullet }{)}^{\wedge }\to (D^{\bullet }{)}^{\wedge }$ is a weak equivalence, where $\wedge$ here means levelwise classical $(p)$-completion.

Consider the following double chain complex:

which we think of as the underlying double chain complex of a cosimplicial cdga, in which every morphism is a quasi-isomorphism of cdgas by virtue of Computation 6.9. Each row except the first row is quasi-isomorphic to zero by [5] Lemma 14.4.1 and [11] tag 07KQ, so by considering the spectral sequence associated to this double complex we see that the totalization of the underlying cosimplicial $E_{\infty }$-ring of this cosimplicial cdga is equivalent to the totalization of the first row when being regarded as a cosimplicial ring. Similar things hold when we levelwise modulo $p$, so similar things hold when we take levelwise classical $(p)$-completion by derived Nakayama lemma since everything in the above double complex is $p$-torsion free. Summarizing the above discussion we see that the totalization of the cosimplicial ring $D_{J_{\bullet }}(P^{\bullet }{)}^{\wedge }$ in the $\infty$-category of $E_{\infty }$-ring is equivalent to the underlying $E_{\infty }$-ring of the cdga $(D_{J_0}(P^0){\otimes }_{P^0}^d{\Omega }_{P^0/{\mathbb{Z}}_p}^{\bullet }{)}^{\wedge }\cong ({\Omega }_{P^0/{\mathbb{Z}}_p}^{\bullet }{)}^{\wedge }$. Moreover, under the complicated chain of equivalences $(C^{\bullet }{)}^{\wedge }\simeq ({\Omega }_{P^0/{\mathbb{Z}}_p}^{\bullet }{)}^{\wedge }$ obtained by summarizing all the efforts we’ve made above, the map $R\to {\pi }_0((C^{\bullet }{)}^{\wedge }{\otimes }_A\overline{A})$ corresponds to the map $R\to {\pi }_0(({\Omega }_{P^0/{\mathbb{Z}}_p}^{\bullet }{)}^{\wedge }{\otimes }_A\overline{A})$, mapping each free variable $x_i\in R$ ($1\le i\le n$) to the element $x^p\in P^0/p{P}^0=({\Omega }_{P^0/{\mathbb{Z}}_p}^0{)}^{\wedge }{\otimes }_A\overline{A}$. Using the Cartier isomorphism (see [5] Lemma 14.1.6) we see that the Hodge-Tate comparison theorem holds in this case.

Lemma 6.11. Let $C$ be an ordinary ring. Let $J\subseteq C$ be a finitely generated ideal. Let $C\to C^{\prime }$ be a map of ordinary rings that has finite $J$-complete Tor amplitude. Then the functor $(-{\otimes }_C{C}^{\prime }{)}^{\wedge }:{\mathcal{D}}^{\le 0}(C)\to \mathcal{D}(C^{\prime })$ preserves totalizations (i.e. limits of cosimplicial objects), where $({)}^{\wedge }$ means nondiscrete derived $J$-completion.

Proof. See [3] Lemma 4.22. Notice that we only need to consider limits of semicosimplicial objects.

$\square$

Construction 6.12 (Functoriality of Prismatic Cohomology). Let $(A,I)$ be a prism. Let $R$, $S$ be two ordinary $\overline{A}$-algebras and let $R\to S$ be a map of $\overline{A}$-algebras. We have a natural restriction functor $(S/A{)}_{\mathbf{\Delta }}\to (R/A{)}_{\mathbf{\Delta }}$ and consequently a sequence of pullback functors $E_{\infty }\text{-Ring}\to {\mathrm{Sh}}_{E_{\infty }\text{-Ring}}((R/A{)}_{\mathbf{\Delta }})\to {\mathrm{Sh}}_{E_{\infty }\text{-Ring}}((S/A{)}_{\mathbf{\Delta }})$ in which every functor admits a right adjoint by adjoint functor theorem. Using adjointness we see that we have a natural commutative diagram of $E_{\infty }$-rings:

Proposition 6.13 (Etale Localization for Hodge-Tate Cohomology). Let $(A,I)$ be a bounded prism. Let $R\to S$ be a $(p)$-completely etale map of ordinary $(p)$-completely smooth $\overline{A}$-algebras. Then the natural map ${\overline{\mathbf{\Delta }}}_{R/A}{\otimes }_{R}S\to {\overline{\mathbf{\Delta }}}_{S/A}$ exhibits ${\overline{\mathbf{\Delta }}}_{S/A}$ as the nondiscrete derived $(p)$-completion of ${\overline{\mathbf{\Delta }}}_{R/A}{\otimes }_{R}S$, when regarding both objects as $S$-module spectra.

Proof. We first prove that the natural functor $F:(S/A{)}_{\mathbf{\Delta }}^{\mathrm{op}}\to (R/A{)}_{\mathbf{\Delta }}^{\mathrm{op}}$ admits a left adjoint $G$. Use Elkik’s Algebraization theorem to choose an ordinary etale $R$-algebra $S^{\prime }$ and a map $S^{\prime }\to S$ of $R$-algebras such that $S$ is the nondiscrete derived $(p)$-completion of $S^{\prime }$. Consider an object

of $(R/A{)}_{\mathbf{\Delta }}^{\mathrm{op}}$. Let $C_0=B/IB{\otimes }_B^{d}S$. Use [11] tag 04D1 to choose an ordinary etale $B$-algebra $C^{\prime }$ such that $C^{\prime }/I{C}^{\prime }\simeq C_0$ as ordinary $B/IB$-algebras. Let $C$ be the discrete derived $(p,I)$-completion of $C^{\prime }$, so it admits a natural $\delta$-ring structure by Proposition 2.4. Let $(D,ID)$ be the prismatic envelope of $(C,IC)$ relative to $(A,IA)$. We claim that $(D,ID)$ is the $G((B,IB))$ we are searching for. Let

be an object of $(S/A{)}_{\mathbf{\Delta }}^{\mathrm{op}}$, equipped with a map $(B,IB)\to F((E,IE))$ in $(R/A{)}_{\mathbf{\Delta }}^{\mathrm{op}}$. Unpacking the definitions, we see that the claim results from the fact that the lifting problem

admits a unique solution, since $B\to C^{\prime }$ is etale and $E\to E/IE$ is a transfinite composition of square zero extensions of animated rings, as $E$ is derived $I$-complete (this could be seen by writing $E$ as the limit of koszul complexes).

Notice that if in the above process the prism $(B,IB)$ is bounded, then $C$ is a $(p,I)$-completely etale $\delta$-$B$-algebra by Proposition 3.5, so $(C,IC)$ is a bounded prism by Proposition 3.7. In particular there is no need to produce the prismatic envelope $(D,ID)$ of $(C,IC)$ in the above construction. In this situation we have a commutative diagram

and it is easy to see that the induced map $B/IB{\otimes }_{R}S\to C/IC$ exhibits $C/IC$ as the nondiscrete derived $(p)$-completion of $B/IB{\otimes }_{R}S$.

Now choose a weakly final object of $(R/A{)}_{\mathbf{\Delta }}$ whose Cech nerve consists of bounded prisms (such an object exists by 4.12). Applying the functor $G^{\mathrm{op}}$ to this object and its Cech nerve we obtain a weakly final object of $(S/A{)}_{\mathbf{\Delta }}$ satisfying the same conditions. We can use these two Cech nerves to compute ${\overline{\mathbf{\Delta }}}_{R/A}$ and ${\overline{\mathbf{\Delta }}}_{S/A}$, and thus the proposition follows from Lemma 6.11 since $R\to S$ is $(p)$-completely flat and consequently has finite $(p)$-complete Tor amplitude.

$\square$

6.14. Let $(A,I)$ be a bounded prism. Let $R\to S$ be a $(p)$-completely etale map of ordinary $(p)$-completely smooth $\overline{A}$-algebras. Construction 5.6 is functorial in $M$, we therefore have a commutative diagram

for every $i\in {\mathbb{Z}}_{\ge 0}$, where the left and right vertical arrows are the maps appearing in the Hodge-Tate comparison map, and are in particular maps of discrete $R$ and $S$ modules, respectively.

Since ${\hat{\Omega }}_{R/\overline{A}}^i$ is a finite projective discrete $R$-module for every $i\in {\mathbb{Z}}_{\ge 0}$ (Computation 5.5), we can choose a map ${⨁}_{i\in {\mathbb{Z}}_{\ge 0}}({\hat{\Omega }}_{R/\overline{A}}^i\{-i\})[-i]\to {\overline{\mathbf{\Delta }}}_{R/A}$ of $R$-module spectra that induces the correct maps ${\hat{\Omega }}_{R/\overline{A}}^i\to {\pi }_{-i}({\overline{\mathbf{\Delta }}}_{R/A})\{i\}$ on homotopy groups (that is, the maps appearing in the Hodge-Tate comparison map) for each $i\in {\mathbb{Z}}_{\ge 0}$ (notice that ${\hat{\Omega }}_{R/\overline{A}}^i\{-i\}:={\hat{\Omega }}_{R/\overline{A}}^i{\otimes }_{\overline{A}}^{d}I/I^2$ has a natural structure of a discrete left $R$-module, which is finite projective). This property remains valid after derived base change along $R\to S$ and then taking nondiscrete derived $(p)$-completion (this statement makes sense in view of Proposition 6.13 and the fact that the nondiscrete derived $(p)$-completion of the base change of ${⨁}_{i\in {\mathbb{Z}}_{\ge 0}}({\hat{\Omega }}_{R/\overline{A}}^i\{-i\})[-i]$ along $R\to S$ is ${⨁}_{i\in {\mathbb{Z}}_{\ge 0}}({\hat{\Omega }}_{S/\overline{A}}^i\{-i\})[-i]$). In particular we have the following

Corollary 6.15. Let $(A,I)$ be a bounded prism. Let $R\to S$ be a $(p)$-completely etale map of ordinary $(p)$-completely smooth $\overline{A}$-algebras. If the Hodge-Tate comparison theorem holds for $R/A$, then it holds for $S/A$.

Conversely, Let $(A,I)$ be a bounded prism. Let $R$ be an ordinary smooth $A$-algebra, let $R_1,...,R_t$ be a finite collection of ordinary etale $R$-algebras such that $R\to {\prod }_{1\le i\le t}{R}_i$ is faithfully flat. Let $R^{\prime }$ and $R_1^{\prime },...,R_t^{\prime }$ be the nondiscrete derived $(p)$-completions of $R$ and $R_1,...,R_t$, respectively (which are infact discrete by Proposition 3.5). If the Hodge-Tate comparison theorem holds for $R_1^{\prime }/A,...,R_t^{\prime }/A$, then it holds for $R^{\prime }/A$.

Construction 6.16 (Base Change for Prismatic Cohomology). Let $(A,I)\to (A^{\prime },I{A}^{\prime })$ be a map of bounded prisms. Let $R$ be an ordinary $(p)$-completely smooth $\overline{A}$-algebra. Let $R^{\prime }=(R{\otimes }_{\overline{A}}\overline{A^{\prime }}{)}^{\wedge }$ (where $({)}^{\wedge }$ means nondiscrete derived $(p)$-completion). $R^{\prime }$ is $(p)$-completely flat over $\overline{A^{\prime }}$ so it is discrete by [2] Lemma 4.7. Moreover $R^{\prime }$ is $(p)$-completely smooth over $\overline{A^{\prime }}$.

There is a natural functor $(R^{\prime }/A^{\prime }{)}_{\mathbf{\Delta }}\to (R/A{)}_{\mathbf{\Delta }}$. Similar to the things we’ve done in Construction 6.12 we obtain a commutative diagram of $E_{\infty }$-rings

Proposition 6.17 (Base Change for Prismatic and Hodge-Tate Cohomology). Let $(A,I)\to (A^{\prime },I{A}^{\prime })$ be a map of bounded prisms such that the map $A\to A^{\prime }$ has finite $(p,I)$-complete Tor amplitude. Let $R$ be an ordinary $(p)$-completely smooth $\overline{A}$-algebra. Let $R^{\prime }=(R{\otimes }_{\overline{A}}\overline{A^{\prime }}{)}^{\wedge }$ (where $({)}^{\wedge }$ means nondiscrete derived $(p)$-completion). Then the natural map ${\mathbf{\Delta }}_{R/A}{\otimes }_A{A}^{\prime }\to {\mathbf{\Delta }}_{R^{\prime }/A^{\prime }}$ exhibits ${\mathbf{\Delta }}_{R^{\prime }/A^{\prime }}$ as the nondiscrete derived $(p,I)$-completion of ${\mathbf{\Delta }}_{R/A}{\otimes }_A{A}^{\prime }$, the natural map ${\overline{\mathbf{\Delta }}}_{R/A}{\otimes }_A{A}^{\prime }\to {\overline{\mathbf{\Delta }}}_{R^{\prime }/A^{\prime }}$ exhibits ${\overline{\mathbf{\Delta }}}_{R^{\prime }/A^{\prime }}$ as the nondiscrete derived $(p,I)$-completion of ${\overline{\mathbf{\Delta }}}_{R/A}{\otimes }_A{A}^{\prime }$.

Proof. Let $R^{\prime \prime }=R{\otimes }_{\overline{A}}^d\overline{A^{\prime }}$, then there is a natural isomorphism of categories $(R^{\prime }/A^{\prime }{)}_{\mathbf{\Delta }}\to (R^{\prime \prime }/A^{\prime }{)}_{\mathbf{\Delta }}$, so we may work with $R^{\prime \prime }$ instead of $R^{\prime }$. The construction of Proposition 4.8 yields a weakly final object of $(R/A{)}_{\mathbf{\Delta }}$, and this object and its Cech nerve commutes with base change along $A\to A^{\prime }$, in the sense of Proposition 4.10 (ii). Therefore if we apply termwise nondiscrete derived $(p,I)$-completion to the cosimplicial ring yield by the cech nerve of this object (when computing prismatic or Hodge-Tate cohomology), we obtain the cosimplicial ring yield by the cech nerve after base change. We can now conclude using Lemma 6.11.

$\square$

6.18. Let $(A,I)\to (A^{\prime },I{A}^{\prime })$ be a map of bounded prisms. Let $R$ be an ordinary $(p)$-completely smooth $\overline{A}$-algebra. Let $R^{\prime }=(R{\otimes }_{\overline{A}}\overline{A^{\prime }}{)}^{\wedge }$ (where $({)}^{\wedge }$ means nondiscrete derived $(p)$-completion). $R^{\prime }$ is $(p)$-completely flat over $\overline{A^{\prime }}$ so it is discrete by [2] Lemma 4.7. Moreover $R^{\prime }$ is $(p)$-completely smooth over $\overline{A^{\prime }}$.

Similar construction as in 6.14 applies, and we have the following

Corollary 6.19. Let $(A,I)\to (A^{\prime },I{A}^{\prime })$ be a map of bounded prisms such that the map $A\to A^{\prime }$ has finite $(p,I)$-complete Tor amplitude. Let $R$ be an ordinary $(p)$-completely smooth $\overline{A}$-algebra. Let $R^{\prime }=(R{\otimes }_{\overline{A}}\overline{A^{\prime }}{)}^{\wedge }$ (where $({)}^{\wedge }$ means nondiscrete derived $(p)$-completion). If the Hodge-Tate comparison theorem holds for $R/A$, then it holds for $R^{\prime }/A^{\prime }$.

Conversely, suppose the functor ${\mathcal{D}}_{\text{comp}}(A)\to {\mathcal{D}}_{\text{comp}}(A^{\prime })$ (where ${\mathcal{D}}_{\text{comp}}(A)$ means the full subcategory of $\mathcal{D}(A)$ spanned by those derived $(p,I)$-complete $A$-module spectra, and similarly for ${\mathcal{D}}_{\text{comp}}(A^{\prime })$) given by first derived base change along $A\to A^{\prime }$ then taking nondiscrete derived $(p,I)$-completion is conservative. Then if the Hodge-Tate comparison theorem holds for $R^{\prime }/A^{\prime }$, then it holds for $R/A$.

Corollary 6.20 (Hodge-Tate Comparison Theorem for Crystalline Prisms). In the situation of Theorem 5.8, if $(A,I)$ is crystalline, then the Hodge-Tate comparison theorem holds.

Proof. Using [11] tag 054L and Corollary 6.15 we may reduce to the case where $R$ is the nondiscrete derived $(p)$-completion of a finite polynomial $\overline{A}$-algebra. We may then use Computation 6.10 and Corollary 6.19 to conclude.

$\square$

Construction 6.21 (The Universal Oriented Prism). Let $A_0={\mathbb{Z}}_{(p)}\{d\}$ be the free $\delta$-ring in a single variable $d$ over ${\mathbb{Z}}_{(p)}$. Let $S$ be the multiplicative subset of $A_0$ generated by ${\varphi }^n(\delta (d))$ for all $n\in {\mathbb{Z}}_{\ge 0}$. By [5] Lemma 2.4.8 the localization $A_1=S^{-1}{A}_0$ admits a natural structure of a $\delta$-ring. Let $A$ be the nondiscrete derived $(p,d)$-completion of $A_1$, which is discrete and classically $(p,d)$-complete by 2.3. It is easy to check that $(A,dA)$ is a bounded prism. It can be checked that $(A,dA)$ has the following property: If $(B,J)$ is an orientable prism, then maps of prisms from $(A,I)$ to $(B,J)$ are in bijection with orientations of $(B,J)$, functorial in $(B,J)$.

As $A/pA$ is $d$-torsion free, one can check that $A/pA{\otimes }_{\varphi ,A/pA}A/(p,d)\simeq A/(p,d^p)$, so $\varphi :A/pA\to A/pA$ is $(d)$-completely flat. Let $B$ be the discrete derived $(p)$-completion of $A\{\frac{\varphi (d)}{p}\}$ (see Notation 6.6). The same proof of Proposition 6.7 shows that $A\{\frac{\varphi (d)}{p}\}$ is $p$-torsion free, so 2.3 applies and $B$ is classically $(p)$-complete. Since $d^p=p(\varphi (d)/p-\delta (d))$ in $B$ we see that $B$ is classically $(p,d)$-complete. Furthermore $B$ equals the classical $(p)$-completion of the divided power envelope of $(A,(d))$ by the proof of Proposition 6.7. $\delta (\varphi (d))=\varphi (\delta (d))$ in $B$ so $\varphi (d)\in B$ is a distinguished element. Since $p$ is also a distinguished element of $B$ and $\varphi (d)$ is divisible by $p$ we may use [5] Lemma 5.2.1 to conclude that $\varphi (d)$ and $p$ generate the same ideal in $B$. Therefore $(B,pB)$ is a crystalline prism and the composition map $A\stackrel{\varphi }{\to }A\to B$, being equal to $A\to B\stackrel{\varphi }{\to }B$, can be promoted to a map $(A,(d))\to (B,(d))\stackrel{\varphi }{\to }(B,(p))$ of $\delta$-pairs, and we obtain a map $(A,(d))\to (B,(p))$ of prisms. Modulo $p$, this map factors as $A/p\to A/(p,d)\stackrel{\varphi }{\to }A/(p,d^p)\to B/p$, where the first map has finite $(d)$-complete Tor amplitude since $A/p$ is $d$-torsion free, the second map is faithfully flat by construction, and the third map is faithfully flat by the structure of divided power envelope. So we see that $A\to B$ has finite $(p,d)$-complete Tor amplitude and satisfies the condition in the second paragraph of Corollary 6.19 (see [3] Construction 6.1 for more details). Summarizing, and using Corollary 6.19 we obtain the following

Corollary 6.22 (Hodge-Tate Comparison Theorem for the Universal Oriented Prism). Let $(A,(d))$ be the universal oriented prism constructed above, let $R$ be an ordinary $(p)$-completely smooth $\overline{A}$-algebra. Then the Hodge-Tate comparison theorem holds.

Theorem 6.23 (Hodge-Tate Comparison Theorem). Let (A,I) be a bounded prism, let $R$ be an ordinary $(p)$-completely smooth $\overline{A}$-algebra. Then the Hodge-Tate comparison theorem holds.

Proof. Using [11] tag 054L and Corollary 6.15 we may reduce to the case where $R$ is the nondiscrete derived $(p)$-completion of a finite polynomial $\overline{A}$-algebra. Using Corollary 6.19 and [3] Lemma 3.7 (4) we may reduce to the case where $(A,I)$ is orientable. Let $(A_0,(d_0))$ denote the universal oriented prism constructed in Construction 6.21. Choose a generator $d$ for $I$, so that we have a map of prisms $(A_0,(d_0))\to (A,I)$ mapping $d_0$ to $d$. Let $(B,(p))$ be the crystalline prism as the one denoted by the same letter in Construction 6.21. The rest of the proof proceed just as in the third paragraph of the proof of [3] Proposition 6.2, which I’m too lazy to copy down.

$\square$

Remark 6.24. For those who are worried of using the unproved claim 5.1 to prove Proposition 5.7, there is a functorial approach, see [3] Proposition 6.2.

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